Dynamic Nuclear Polarization and the Paradox of Quantum Thermalization

Dynamic nuclear polarization (DNP) is to date the most effective technique to increase the nuclear polarization opening disruptive perspectives for medical applications. In a DNP setting, the interacting spin system is quasi-isolated and brought out of equilibrium by microwave irradiation. Here we show that the resulting stationary state strongly depends on the ergodicity properties of the spin many-body eigenstates. In particular, the dipolar interactions compete with the disorder induced by local magnetic fields resulting in two distinct dynamical phases: while for weak interaction, only a small enhancement of polarization is observed, for strong interactions the spins collectively equilibrate to an extremely low effective temperature that boosts DNP efficiency. We argue that these two phases are intimately related to the problem of thermalization in closed quantum systems where a many-body localization transition can occur varying the strength of the interactions.

Figure 1

A solid material containing nuclear spins (e.g., ${}^{13}\mathrm{C}$, ${}^{15}\mathrm{N}$) and doped with molecules with unpaired electrons (left). At 1.2 K and 3.35 T the equilibrium polarization of the electron spins is very high, (94%), while nuclear spins are very little polarized, less than 1%. Under microwave irradiation the spin system evolves towards a new steady state characterized by a single spin temperature ${\beta }_s^{-1}\sim 1\mathrm{mK}$ (right). In this work, we analyze exclusively the electron spins and show that an out-of-equilibrium spin temperature results from the interplay of disorder and interaction.

Figure 2

EPR spectrum. Equilibrium (blue) versus MW irradiated (yellow) profile. Under irradiation two possible profiles are expected: (a) The hole burning shape, characteristic of the noninteracting case; (b) the hyperbolic tangent shape characterized by a very low effective temperature ${\beta }_s$ that cools down the nuclear spins.

Figure 3

Electron polarization under MW irradiation with ${\omega }_{\mathrm{MW}}=93.8775\mathrm{GHz}$, for the model of Eq. (1) with $N=12$ spins. For strong dipolar interactions (circles and diamonds) the spin temperature is well defined. The fit according to Eq. (2) (red and green lines) gives ${\beta }_s^{-1}=3.7\mathrm{mK}$ for $U=15\mathrm{MHz}$ and ${\beta }_s^{-1}=7.4\mathrm{mK}$ for $U=45\mathrm{MHz}$. For weak interactions $U=2\mathrm{MHz}$ (square) a simple broadening of the hole burning noninteracting profile (dashed blue line) given in Eq. (8) of Ref. [19].

Figure 4

Left: Density plot of the distribution of diagonal elements $⟨n|{\widehat{S}}_z^i|n⟩$ in the sector of vanishing total polarization. Colored regions represent, for each energy window $(\epsilon ,\epsilon +\mathrm{\Delta }\epsilon )$, the smallest area containing half probability ($U=2\mathrm{MHz}$ in blue, $U=15\mathrm{MHz}$ in red, $U=45\mathrm{MHz}$ in green). Middle: Logarithm of the variation of the local polarization between pairs of adjacent eigenstates of Eq. (1) versus $N$. In the ergodic phase, this indicator vanishes exponentially in $N$. In the localized phase, it saturates to a finite value. Right: EPR spectrum for the toy model of Eq. (5) with 7 packets. The equilibrium profile is in blue. The yellow histograms show the stationary profile under MW irradiation with ${\omega }_{\mathrm{MW}}=93.8775\mathrm{GHz}$, $N=64$ spins and $f_p=N_p/N$. The solid line is obtained from the ansatz of Eq. (2) imposing the condition (6).

Figure 5

$T_1$ shortening effect. The spin temperature ${\beta }_s$, obtained from the fit of the electron polarizations (see Fig. 3), is shown versus ${\omega }_{\mathrm{MW}}$ with $U=15\mathrm{MHz}$ and different values of the relaxation time $T_1$: the spin temperature is smaller when relaxation is faster, consistently with the observed increase in the hyperpolarization efficiency [34, 35].